On spaces of analytic functions of several variables (Q2759399)

This paper deals with topological isomorphism between some locally convex spaces of holomorphic or entire functions of several complex variables and abstract analytic spaces. NEWLINENEWLINENEWLINELet the series \(\sum_{k\in\mathbb{Z}_{+}^{n}}f_{k}z^{k}=\sum_{k\in K_{f}}f_{k}z^{k_1}_1\ldots z_{n}^{k_{n}}\) \((K_{f}=\{k\in\mathbb{Z}_{+}^{n}:f_{k}\neq 0\})\) have a non-empty convergence domain \(D(f)\subset\mathbb{C}^{n}\), i.e. represents the function \(f=(f_{k})\) holomorphic in \(0\in\mathbb{C}^{n}\). We denote by \(\overline A_0\) the set of all functions \(f\) holomorphic in \(0\). Let \(\Phi=\{\Phi_{k}:\mathbb{R}_{+}\to\overline\mathbb{R}_{+}, k\in\mathbb{Z}_{+}^{n}\}\) be a characteristic, and let \(\chi_{\Phi}^{f}:\Sigma_{+}\to \overline\mathbb{R}_{+}\) be directed \(\Phi\)-characteristic, \(\chi_{\Phi}^{f}(a)=\lim_{\pi(k)\to a}\Phi_{k}(|f_{k}|)\), if \(a\in \Sigma_{K_{f}}\), and \(\chi_{\Phi}^{f}(a)=-\infty\), if \(a\in \Sigma_{+}\setminus\Sigma_{K_{f}}\), \(\pi(k)=k/\|k\|, k\in\mathbb{Z}_{+}^{n}\). The hypersurface \(S_{f}\) of conjugate \((\Phi,\Psi)\)-characteristics of the function \(f\) is defined by NEWLINE\[NEWLINES_{f}=\Bigl\{r\in D_{\Psi}:\max_{a\in\Sigma_{+}}\chi_{\Phi}^{f}(a)\nu(\langle\beta(a),\eta^{-1}(r)\rangle)=1\Bigr\}=\partial[\eta(H_{\Phi}^{f}\cap\Delta_{\Psi})],NEWLINE\]NEWLINE where \(\eta\) is a homeomorphism of the convex domain \(\Delta_{\Psi}\subset\mathbb{R}^{n}\) on the domain \(D_{\Psi}\subset\mathbb{R}^{n}\); \(\nu:]a,b[\to \widehat\mathbb{R}\) and \(\beta:\Sigma_{+}\to\mathbb{R}^{n}\setminus\{0\}\) are homeomorphic mappings such that \(\langle\pi(\beta(a)),a\rangle>0,\;a\in \Sigma_{+}\), \(\inf\{\langle\beta(a),\xi\rangle:\xi\in\Delta_{\Psi}\}=a\), \(\sup\{\langle\beta(a),\xi\rangle:\xi\in\Delta_{\Psi}\}=b\), \(H_{\Phi}^{f}:=\bigcap_{0<\chi_{\Phi}^{f}(a)<\infty}\{\xi\in\mathbb{R}^{n}:\nu(\langle\beta(a),\xi\rangle)\leq 1/\chi_{\Phi}^{f}(a)\}\). Let us continue \(\nu\) to the homeomorphism \(\tilde\nu:[a,b]\to\overline\mathbb{R}^{n}_{+}\). Let \(\gamma\) be a fixed function from \(\overline A_0\) and let \(\Omega\) be the interior of the set \(\eta(H_{\Phi}^{\gamma}\cap\Delta_{\Psi})\), \(\Omega\neq\emptyset\) and \(\Omega\neq\eta(\Delta_{\Psi})=D_{\Psi}\). Let us denote by \(h\) the support function of the convex set \(\eta^{-1}(\Omega)\), and let us introduce subspaces \(A_{\Phi}[\Omega]\) and \(A_{\Phi}[\Omega)\) of the vector space \(\overline A_0\) in such manner: \(f\in A_{\Phi}[\Omega]\), (\(f\in A_{\Phi}[\Omega)\)) if and only if \(\chi_{\Phi}^{f}\leq 0\) or \(\chi_{\Phi}^{f}<\infty\) and \(S_{f}\cap\Omega=\emptyset\) (\(\chi_{\Phi}^{f}<\infty\) and \(d(S_{f},\Omega)>0\), where \(d\) is the Euclidean metric. NEWLINENEWLINENEWLINEThe author proves that the vector spaces \(A_{\Phi}[\Omega]\) and \(A_{\Phi}[\Omega)\) are isomorphic respectively to abstract analytic spaces \(A_{\Phi}(\varphi)\) and \(\overline{A}_{\Phi}(\varphi)\), where \(\varphi(a)=1/\widetilde\nu(\varepsilon h(\varepsilon\beta(a))), a\in\Sigma_{+}\), \(\varepsilon=+1\) or \(\varepsilon=-1\) depending on either function \(\nu\) is increasing or decreasing respectively. If \(\Omega=D_{\Psi}\), then \(\varphi(a)\equiv 0\), if \(\Omega=\emptyset\), then \(\varphi(a)\equiv+\infty\). The author also proves that spaces \(A_{\Phi}[\Omega]\) and \(A_{\Phi}[\Omega)\) are inductive and projective limit of a sequence of Banach spaces respectively. Examples of the corresponding spaces and characteristics are presented.